By the construction of products in \(\sigma [M]\)  (see [1]}), we know that \(\prod _{\lambda \in \Lambda }^{M}Y_{\lambda }\)  together with the restrictions \(\overline{\pi }_{\lambda }\)  of the canonical projections \(\pi _{\lambda }:\prod _{\lambda \in \Lambda }Y_{\lambda }\rightarrow Y_{\lambda } \)  is the product in \(\sigma [M]\) . Then we can define two group homomorphisms \(\phi :\mbox{Hom}_{\sigma [M]}(X,\prod _{\lambda \in \Lambda }^{M}Y_{\lambda })\rightarrow \prod _{\lambda \in \Lambda }\mbox{Hom}_{\sigma [M]}(X,Y_{\lambda })\)  and \(\Phi :\mbox{Hom}_{R}(X,\prod _{\lambda \in \Lambda }Y_{\lambda })\rightarrow \prod _{\lambda \in \Lambda }\mbox{Hom}_{R}(X,Y_{\lambda })\) , which are given by \(\phi (f)=\lbrace f\overline{\pi }_{\lambda }\rbrace _{\lambda \in \Lambda }\)  and \(\Phi (g)=\lbrace g\pi _{\lambda }\rbrace _{\lambda \in \Lambda }\)  for any \(f\in \mbox{Hom}_{\sigma [M]}(X,\prod _{\lambda \in \Lambda }^{M}Y_{\lambda })\) , \(g\in \mbox{Hom}_{R}(X,\prod _{\lambda \in \Lambda }Y_{\lambda })\) . It is well-known that \(\Phi \)  is an isomorphism. Consider the following commutative diagram
\({0[r]&\mbox{Hom}_{\sigma [M]}(X,\prod _{\lambda \in \Lambda }^{M}Y_{\lambda })[r]^{inc}[d]^{\phi } & \mbox{Hom}_{R}(X,\prod _{\lambda \in \Lambda }Y_{\lambda })[d]^{\Phi }\\0[r]&\prod _{\lambda \in \Lambda }\mbox{Hom}_{\sigma [M]}(X,Y_{\lambda })[r]^{id} & \prod _{\lambda \in \Lambda }\mbox{Hom}_{R}(X,Y_{\lambda }) }\) 
where \(inc\)  is a inclusion map and \(id\)  is an identity. It implies that \(\phi \)  is injective.
